Finite Group DatabasebetaFinite Group Databasebeta| order | 5312 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | C22⋊C8×C83 ⋊ C2 |
| Sylow | C83 ⋊ #407 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2×C4 | C22×D83 |
| commutator, socle | C22×C83 | C22×C4 |
| Fitting | C22⋊C8×C83 | C2 |
| hypercenter | C22⋊C8 | D83 |
| hypocenter | C83 | #407 |
| LCS | G ▷ C22×C83 ▷ C83 | |
| UCS | C1 ◁ C2×C4 ◁ C22⋊C8 | |
190 conjugacy classes of subgroups; only the 61 normal subgroups are shown.
This group has 190 conjugacy classes of subgroups — too many to draw. Only its 61 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
Polycyclic presentation:
2^6,83|a^2=e,e^2=f,b^a=bd,c^a=cf,g^c=g^82848 conjugacy classes — view the full character table.
| maximal subgroup of | #407×D83, C2×#19127663, #24915905, … |
| maximal normal subgroup of | #407×D83, C2×#19127663, #24915905, … |
| maximal quotient of | #24907155, #24907224, #24907227, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(5312, 511);